Wild models of curves

Let $K$ be a complete discrete valuation field with ring of integers $O_{K}$ and algebraically closed residue field $k$ of characteristic $p > 0$ . Let $X ∕ K$ be a smooth proper geometrically connected curve of genus $g > 0$ with $X (K) \neq \emptyset$ if $g = 1$ . Assume that $X ∕ K$ does not have good reduction and that it obtains good reduction over a Galois extension $L ∕ K$ of degree $p$ . Let $Y ∕ O_{L}$ be the smooth model of $X_{L} ∕ L$ . Let $H : = Gal (L ∕ K)$ .

In this article, we provide information on the regular model of $X ∕ K$ obtained by desingularizing the wild quotient singularities of the quotient $Y ∕ H$ . The most precise information on the resolution of these quotient singularities is obtained when the special fiber $Y_{k} ∕ k$ is ordinary. As a corollary, we are able to produce for each odd prime $p$ an infinite class of wild quotient singularities having pairwise distinct resolution graphs. The information on the regular model of $X ∕ K$ also allows us to gather insight into the $p$ -part of the component group of the Néron model of the Jacobian of $X$ .

Keywords

model of a curve, ordinary curve, cyclic quotient singularity, wild ramification, arithmetical tree, resolution graph, component group, Néron model

Mathematical Subject Classification 2010

Primary: 14G20

Secondary: 14G17, 14K15, 14J17

Milestones

Received: 3 January 2013

Revised: 6 June 2013

Accepted: 16 July 2013

Published: 18 May 2014

Authors

Dino Lorenzini
Department of Mathematics University of Georgia Athens, GA 30602 United States

Vol. 8, No. 2, 2014

Dino Lorenzini

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors